Isochronous n -dimensional nonlinear PDM-oscillators: linearizability, invariance and exact solvability
aa r X i v : . [ n li n . S I] A ug Isochronous n -dimensional nonlinear PDM-oscillators: linearizability, exact solvabilityand ˙ H -invariance Omar Mustafa ∗ Department of Physics, Eastern Mediterranean University, G. Magusa, north Cyprus, Mersin 10 - Turkey,Tel.: +90 392 6301378; fax: +90 3692 365 1604.
Abstract:
Within the standard Lagrangian settings (i.e., the difference between kinetic and po-tential energies), we discuss and report isochronicity, linearizability and exact solvability of some n -dimensional nonlinear position-dependent mass (PDM) oscillators. In the process, negative thegradient of the PDM-potential force field is shown to be no longer related to the time derivativeof the canonical momentum, p = m ( r ) ˙r , but it is rather related to the time derivative of thepseudo-momentum, π ( r ) = p m ( r ) ˙r (i.e., Noether momentum). Moreover, using some point trans-formation recipe, we show that the linearizability of the n -dimensional nonlinear PDM-oscillators isonly possible for n = 1 but not for n ≥
2. The Euler-Lagrange invariance falls short/incomplete for n ≥ n -dimensional PDM ˙ H -invariance (i.e., time derivative of the Hamiltonian). Such invariance, inaddition to Newtonian invariance of Mustafa [42] authorizes, in effect, the use of the exact solutionsof one system to find the solutions of the other. A sample of isochronous n -dimensional nonlinearPDM-oscillators examples are reported. Keywords:
PDM-Lagrangians, PDM nonlinear oscillators, linearizability, isochronicity, ˙ H -invariance, exact solvability. PACS numbers : I. INTRODUCTION
A standard Lagrangian is the difference between kinetic and potential energies (otherwise the Lagrangian is a non-standard one). Likewise, the sum of the kinetic and potential energies represents a standard Hamiltonian, otherwisenon-standard. In fact, such standard presentation for the Lagrangian/Hamiltonian renders the total energy to bean integral of motion (i.e., a constant of motion). However, it should be noted that the most prominent Mathews-Lakshmanan oscillators Lagrangians [1] L ( x, ˙ x ; t ) = 12 (cid:18) ˙ x − ω x ± λx (cid:19) ; ˙ x = dxdt , (1)belongs, obviously, to the set of non-standard Lagrangians. That is, if in the standard textbook harmonic oscillatorLagrangian L ( x, ˙ x ; t ) = m ◦ (cid:0) ˙ x − ω x (cid:1) /
2, the coordinate x is transformed/deformed so that x → p Q ( u ) u , thenthe velocity ˙ x would be transformed/deformed in a completely different manner so that ˙ x → p m ( u ) ˙ u . Where, therelation between the dimensionless scalar functions Q ( u ) and m ( u ) would be determined in the process of enforcing theEuler-Lagrange invariance, or any physically feasible alternative invariance. Such non-standard Mathews-LakshmananLagrangians (1) yield (using Euler-Lagrange equation of motion) the nonlinear dynamical equations¨ x ∓ λx ± λx ˙ x + ω ± λx x = 0 , ¨ x = d xdt , (2)that admit simple harmonic oscillators’ solutions x = A cos (Ω t + ϕ ) ; Ω = ω ± λA . Obviously, the conditions Ω = ω / (cid:0) ± λA (cid:1) suggest that the dynamical equations (2) are conditionally-exactly solv-able. Yet, such conditionally-exact solvability renders the oscillators’ frequencies Ω to be an amplitude-dependent fre-quency. Consequently, the nonlinear oscillators lose their isochronicity and are non-isochronous, therefore. Basically, ifone defines m ( x ) = m ◦ / (cid:0) ± λx (cid:1) , then Lagrangians (1) would (with m ◦ = 1) read L ( x, ˙ x ; t ) = m ( x ) (cid:0) ˙ x − ω x (cid:1) / ∗ Electronic address: [email protected] and may very well be, effectively and metaphorically speaking, classified as position-dependent mass (PDM) La-grangians (but not within the standard Lagrangian settings).Such non-standard Lagrangians’ structure have inspired a great research interest in PDM settings, both in classicaland quantum mechanics (c.f., e.g., the sample of references [1–45]). Moreover, the nonlinear differential form ofthe PDM Euler-Lagrange equations of (2) represents some peculiar cases of the quadratic (i.e., with an ˙ x term)Li´enard-type nonlinear differential equation ¨ x + f ( x ) ˙ x + g ( x ) = 0. (3)Which is, in fact, a very interesting equation for both physics and mathematics [1–11]. The linearizability andisochronicity of which have invited a vast number of interesting research studies in many fields (c.f., e.g., [35–45]).Tiwari et al. [2] and Lakshmanan and Chandrasekar [3], for example, have used Lie point symmetries and assertedthat in the case of eight parameter symmetry group, the one-dimensional quadratic Li´enard type equation (3) islinearizable and isochronic. It should be mentioned, nevertheless, that the Mathews-Lakshmanan oscillators (2) arelinearizable via some nonlocal point transformations [3–5] but not isochronous.In this work, however, we shall be interested in the generalization of such nonlinear PDM-oscillators for any physi-cally viable PDM-settngs. Therefore, we focus our attention on the class of standard PDM Lagrangians/Hamiltoniansand their linearizability that preserves isochronicity (i.e., with amplitude-independent frequencies) in the process (c.f.,e.g., [35–41]). Hereby, isochronous n -dimensional nonlinear PDM-oscillators form the subject of the current method-ical proposal. Consequently, we organize our paper in such a way that the current methodical proposal is made clearand comprehensive to serve for viable/feasible pedagogical implementations of isochronous nonlinear PDM-oscillators.In so doing, we recollect, in section II, some preliminaries on the Mathews-Lakshmanan nonlinear oscillators (1)(within their non-standard Lagrangians/Hamiltonians presentations) so that their generalization to any PDM m ( x )settings is made feasible and safe. We also summarize their exact and conditionally-exact, non-oscillatory and oscilla-tory, feasible solutions. This would allow the reader to clearly figure out the difference between our current standardLagrangians/Hamiltonians proposal and the non-standard Mathews-Lakshmanan nonlinear oscillators (1), along withtheir generalized PDM settings. Within the standard Lagrangian settings, we discuss and report, in section III, thecorrelation between negative the n -dimensional gradient of the PDM potential force field (i.e., the n -dimensional PDMforce) and the pseudo-momentum π ( r ) = p m ( r ) ˙r [5, 7] (i.e., Noether momentum [6]). We show that negative thegradient of the PDM potential force field is no longer the time derivative of the canonical momentum, p = m ( r ) ˙r ,but it is rather related with the time derivative of the pseudo-momentum, π ( r ) = p m ( r ) ˙r (as in (33) below). In thesame section, we introduce a new concept to be called, hereinafter, the n -dimensional PDM ˙ H -invariance . Where,we show that the connection between constant mass settings and PDM settings may very well be established throughsome point transformation (c.f., e.g., [5, 15, 16, 42, 43]). In this case, the Euler-Lagrange invariance is shown to besatisfied in n = 1 dimension but falls short and incomplete for n ≥ H -invariance of the current methodical proposal. Consequently,we discuss the linearizability and ˙ H -invariance of some isochronous n -dimensional PDM oscillators in section IV.Such invariance allows us to use the well known exact solutions of constant mass oscillators and reflect/connect thesesolutions with isochronous nonlinear PDM oscillators. This is illustrated through the two sets of examples in sectionV, where we consider a set of one-dimensional PDM Euler-Lagrangian equations of motion and an n -dimensional one.Therein, isochronicity, linearizability and exact solvability of the samples of nonlinear PDM-oscillators are made clearin step-by-step procedures. Our concluding remarks are given in section VI. II. PRELIMINARIES ON MATHEWS-LAKSHMANAN NONLINEAR PDM-OSCILLATORS:RECOLLECTED AND PDM GENERALIZED
In the generalization of the non-standard Mathews-Lakshmanan oscillators Lagrangian (1) to cover PDM settings,one should keep in mind that Lagrangian (1) is rewritten as L = 12 m ( x ) ˙ x − m ( x ) ω x ; m ( x ) = 11 ± λx , ˙ x = dxdt . (4)This would imply the Euler-Lagrange dynamical system¨ x + m ′ ( x )2 m ( x ) ˙ x + (cid:18) m ′ ( x )2 m ( x ) x (cid:19) ω x = 0; m ′ ( x ) = dm ( x ) dx . (5)Obviously, only under the assumption that (cid:18) m ′ ( x )2 m ( x ) x (cid:19) = m ( x ) , (6)would the PDM function read m ( x ) = 11 ± β x . (7)Which is indeed the PDM used in the Mathews-Lakshmanan oscillator (1), with λ = β for the convenience of thecurrent methodical proposal.However, the linearization of such dynamical system, (5) along with (6), into simple harmonic oscillator d Udτ + ω U = 0 ; U = A cos ( ωτ + ϕ ) (8)may be achieved through two nonlocal transformations (to the best of our knowledge). The first of which (c.f., e.g.,[3, 4]) suggests that U = p m ( x ) x ; dτ = m ( x ) dt ⇐⇒ dUdτ = 1 m ( x ) (cid:18) m ′ ( x )2 m ( x ) x (cid:19) p m ( x ) ˙ x = p m ( x ) ˙ x, (9)and consequently d Udτ = 1 p m ( x ) (cid:18) ¨ x + m ′ ( x )2 m ( x ) ˙ x (cid:19) , (10)to imply that (8) be rewritten as ¨ x + m ′ ( x )2 m ( x ) ˙ x + m ( x ) ω x = 0 . (11)This is a valid result if and only if condition (6) is satisfied to yield the PDM in (7). The second nonlocal transformation(e.g., [5] ) suggests that dU = p g ( x ) dx , dτ = f ( x ) dt ; U = p m ( x ) x. (12)Under such nonlocal transformation setting, we get dUdτ = p g ( x ) f ( x ) ˙ x = (cid:18) m ′ ( x )2 m ( x ) x (cid:19) p m ( x ) f ( x ) ˙ x ⇐⇒ p g ( x ) = (cid:18) m ′ ( x )2 m ( x ) x (cid:19) p m ( x ) . (13)and d Udτ = p g ( x ) f ( x ) (cid:20) ¨ x + (cid:18) g ′ ( x )2 g ( x ) − f ′ ( x ) f ( x ) (cid:19) ˙ x (cid:21) . (14)Hence, equation (8) reads ¨ x + (cid:18) g ′ ( x )2 g ( x ) − f ′ ( x ) f ( x ) (cid:19) ˙ x + f ( x ) p g ( x ) ! p m ( x ) ω x = 0 . (15)Which when compared with (5) implies that g ′ ( x )2 g ( x ) − f ′ ( x ) f ( x ) = m ′ ( x )2 m ( x ) ⇐⇒ p g ( x ) = f ( x ) p m ( x ) ⇐⇒ f ( x ) = (cid:18) m ′ ( x )2 m ( x ) x (cid:19) . (16)As a result, equation (15) collapses into (5). Moreover, if one chooses to work with f ( x ) = m ( x ) then necessarily m ( x ) = 1 / (cid:0) ± β x (cid:1) as in (6) and (7) above. One may conclude that the current PDM generalization of theMathews-Lakshmanan oscillator (1) is now safe and clear. The above PDM settings are, therefore, applicable tosome general PDM m ( x ) in principle (without condition (6) of course), but not within the standard Lagrangianpresentation.Nevertheless, one should be aware that for the PDM in (7) our dynamical equation of (5) (or equivalently, equation(2) with λ = β ) admits non-oscillatory exact solutions in the forms of x ( t ) = ± α β h − (cid:0) α β + ω (cid:1) e − α β ( α + t ) ± e α β ( α + t ) i , (17)and/or x ( t ) = ± α β h − (cid:0) α β + ω (cid:1) e α β ( α + t ) ± e − α β ( α + t ) i . (18)Moreover, a set of conditionally non-oscillatory exact solutions are feasible too. Amongst are x ( t ) = A cosh (Ω t + ϕ ) ; Ω = ∓ ω ± β A , (19)and/or x ( t ) = A sinh (Ω t + ϕ ) ; Ω = ± ω ∓ β A . (20)Yet, the prominent conditionally exact Mathews-Lakshmanan oscillators solutions are given by x ( t ) = A cos (Ω t + ϕ ) ; Ω = ± ω ± β A , (21)and/or x ( t ) = A sin (Ω t + ϕ ) ; Ω = ± ω ± β A (22)It is obvious that the Mathews-Lakshmanan oscillators’ frequencies, in (21) and (22), are amplitude-dependent frequen-cies and are non-isochronous oscillators, therefore. Hereby, we argue that, although the nonlocal point transformationrecipes in (9) and (12) serve to secure invariance between the dynamical systems of (5) and (8), they render theoscillators non-isochronous. This is, in fact, a natural consequence of the position-dependent deformation of the timeelements of (9) and (12) (i.e., dτ = m ( x ) dt in (9) and dτ = f ( x ) dt in (12), where f ( x ) is given in (16)). This wouldnecessarily mean that, if the oscillators isochronicity is the sought-after objective then the time element should notbe a position-dependent deformed one.In what follows, we abort the non-standard Lagrangian presentations and discuss some standard classical mechan-ical textbook Lagrangians under PDM settings. This would be very interesting for any viable/feasible pedagogicalimplementations of PDM Lagrangians/Hamiltonians. III. n -DIMENSIONAL GRADIENT OF PDM-POTENTIAL ENERGY AND PDM ˙ H -INVARIANCE Apiori, it is known that under constant mass setting, the force is the time derivative of the canonical momentumand is given by negative the gradient of the potential force field., i. e., F = d p dt = − ∇ V ( r ) ; ∇ = X j =1 ∂ x j ˆ x j , r = X j =1 x j ˆ x j , F = X j =1 F i ˆ x j , r = vuut X j =1 x j . (23)Under PDM settings, however, negative the gradient of the potential force field is no longer given by the time derivativeof the canonical momentum. In the one-dimensional case, for example, Mustafa [5] has asserted that the relationbetween the force and the potential force field is rather given by F = p m ( x ) ddt (cid:16) m ◦ p m ( x ) ˙ x (cid:17) = − V ′ ( q ( x )) , (24)where V ( q ( x )) is the PDM-deformed potential force field and V ′ ( q ( x )) = dV ( q ( x )) dx ; q ( x ) = Z p m ( x ) dx. Equation (24) is a documentation that, in the one-dimensional case, negative the gradient of the potential force fieldis not equal to the time derivative of the canonical momentum (i.e., dp/dt = − V ′ ( x ) , where p = m ( x ) ˙ x ). So is the n -dimensional case. Consequently, the underlying n -dimensional dynamics of the PDM systems have to be clarifiedin advance. Namely, one has to answer the question as to ”what would negative the gradient of the n -dimensionalPDM-deformed potential force field yield to? That would be the sought-after net PDM force vector. A. Negative the gradient of the PDM potential force field
Hereby, we consider the n -dimensional PDM Lagrangian L ( r , ˙r ; t ) = 12 m ◦ m ( r ) ˙r − V ( q ( r )) = 12 m ◦ m ( r ) n X j =1 ˙ x j − V ( q ( r )) , (25)where q ( r ) = p Q ( r ) r = n X j =1 q j ( r ) ˆ x j (26)and Q ( r ) is some PDM-deformation function manifested by the PDM-deformation m ( r ) of the kinetic energy term .Obviously, the velocity vector ˙r in the kinetic energy term of L ( r , ˙r ; t ) in (25), practically and intuitively, is assumedto be transformed as ˙r −→ p m ( r ) ˙r under PDM settings. It is, therefore, convenient and sufficient to assume thatthe coordinates would transform in a different way as r −→ p Q ( r ) r , as in (26), under the same settings. As long asthe relation between m ( r ) and Q ( r ) is to be determined in the process, this assumption remains valid and sufficient.We now use the Euler-Lagrange equations ddt (cid:18) ∂L∂ ˙ x i (cid:19) − ∂L∂x i = 0; i = 1 , , · · · , n ∈ N , (27)to obtain (with m ◦ = 1 throughout) n PDM Euler-Lagrange equations m ( r ) ¨ x i + ˙ m ( r ) ˙ x i − ∂ x i m ( r ) n X j =1 ˙ x j = − ∂ x i V ( q ( r )) . (28)Next, we multiply each term by ˆ x i and sum over i = 1 , , · · · , n to get the corresponding Newtonian dynamicalequation m ( r ) n X i =1 ¨ x i ˆ x i + ˙ m ( r ) n X i =1 ˙ x i ˆ x i − n X i =1 ∂ x i m ( r ) ˆ x i n X j =1 ˙ x j = − n X i =1 ˆ x i ∂ x i V ( q ( r )) = − ∇ V ( q ( r )) . (29)To avoid mathematical complexities, we may assume that m ( r ) = m ( r ) and Q ( r ) = Q ( r ) where r is readily definedin (23). This would allow us to represent (29) as m ( r ) ¨r + ˙ m ( r ) ˙r − n X i =1 ∂ x i m ( r ) ˆ x i ˙r = − ∇ V ( q ( r )) . (30)However, one may express ˙ m ( r ), with ∂ r m ( ~r ) = ∂m ( ~r ) /∂r , as˙ m ( r ) = n X k =1 ∂ x k m ( r ) ˙ x k = ∂ r m ( r ) r n X k =1 x k ˙ x k = ∂ r m ( r ) r ( r · ˙r ) . (31)and n X i =1 ∂ x i m ( r ) ˆ x i = ∂ r m ( r ) r n X k =1 x k ˆ x k = ∂ r m ( r ) r r = m ′ ( r ) r r , (32)so that equation (30) reads, with r ( ˙r · ˙r ) = ( r · ˙r ) ˙r (i.e., no rotational effects under consideration and r k ˙r ,therefore), m ( r ) ¨r + ˙ m ( r )2 ˙r = − ∇ V ( q ( r )) ⇐⇒ F = p m ( r ) ddt (cid:16)p m ( r ) ˙r (cid:17) = − ∇ V ( q ( r )) . (33)This result would, in fact, represent the n -dimensional PDM Newtonian dynamics. It suggests that in a free forcefield (i.e., V ( q ( r )) = 0), the canonical momentum p = m ( r ) ˙r is no longer a conserved quantity but rather the PDMpseudo-momentum π ( r ) = p m ( r ) ˙r [5, 7] (or in the Cariˇnena et al’s [6] language, the ”Noether momentum”) is theconserved quantity. Moreover, it is now obvious that, under PDM setting, negative the gradient of the potential forcefield is no longer the same as the time derivative of the canonical momentum p . Yet it recovers the constant masssettings for m ( r ) = 1 to yield the usual textbook relation m ◦ ¨r = − ∇ V ( r ). B. n -dimensional PDM ˙ H -invariance Let us consider a standard n -dimensional constant mass Lagrangian L ( q , ˙q ; t ) = 12 m ◦ n X j =1 ˙ q j − V ( q ); ˙ q j = dq j dt ; j = 1 , , · · · , n ∈ N , (34)Then the corresponding n Euler-Lagrange equations (with m ◦ = 1) are given by¨ q i + ∂ q i V ( q ) = 0 ; i = 1 , , · · · , n ∈ N . (35)Under a point transformation in the form of dq i = p m ( r ) dx i ⇐⇒ ∂ x i q i = ∂q i ∂x i = p m ( r ) ⇐⇒ ˙ q i = p m ( r ) ˙ x i ⇐⇒ ˙q = p m ( r ) ˙r , (36)and the assumption that q = p Q ( r ) r ⇐⇒ ˙q = p Q ( r ) (cid:18) Q ′ ( r )2 Q ( r ) r (cid:19) ˙r , (37)the comparison between (36) and (37) would imply that p m ( r ) = p Q ( r ) (cid:18) Q ′ ( r )2 Q ( r ) r (cid:19) . (38)The connection between m ( r ) and Q ( r ) is clear, therefore. We may now proceed with (35) and use˙ q i = p m ( r ) ˙ x i ⇐⇒ ¨ q i = p m ( r ) (cid:20) ¨ x i + ˙ m ( r )2 m ( r ) ˙ x i (cid:21) (39)in (35), along with ∂ q i = ( ∂x i /∂q i ) ∂ x i = m ( r ) − / ∂ x i , to obtain m ( r ) ¨ x i + 12 ˙ m ( r ) ˙ x i + ∂ x i V ( q ) = 0 (40)Which, when compared with (28), suggests that the invariance between (28) and (35) is still far beyond reach at thisstage. However, if we multiply (28) by ˙ x i and sum over i = 1 , , · · · , n we get m ( r ) n X i =1 ˙ x i ¨ x i + ˙ m ( r ) n X i =1 ˙ x i − n X i =1 ˙ x i ∂ x i m ( r ) ! n X j =1 ˙ x j + n X i =1 ˙ x i ∂ x i V ( q ( r )) = 0 , (41)and consequently with ˙ m ( r ) in (31) it reads m ( r ) n X i =1 ˙ x i ¨ x i + 12 ˙ m ( r ) n X i =1 ˙ x i + ˙ V ( q ( r )) = 0 ; ˙ V ( q ( r )) = n X i =1 ˙ x i ∂ x i V ( q ( r )) . (42)Similarly, equation (40) would yield m ( r ) n X i =1 ˙ x i ¨ x i + 12 ˙ m ( r ) n X i =1 ˙ x i + ˙ V ( q ) = 0 (43)Now we got a clear invariance between (42) and (43), through the point transformation of (36). However, a questionof delicate nature arises in the process as to ”what kind of invariance we got here?”Let us rearrange the first two term of either (42) or (43) so that m ( r ) n X i =1 ˙ x i ¨ x i + 12 ˙ m ( r ) n X i =1 ˙ x i = ddt m ( r ) n X i =1 ˙ x i ! . (44)Then equation (42) and (43) are nothings but the time derivative of the total energy of the PDM system. That is,they can both be expressed as ddt n X j =1 ˙ q j − V ( q ) = 0 = ddt (cid:18) p m ( r ) + V ( q ( r )) (cid:19) , (45)where p = m ( r ) ˙r is the canonical momentum. Consequently, one may safely write˙ H ( q , ˙q ; t ) = 0 = ˙ H ( r , ˙r ; t ) . (46)Therefore, we may now conclude that ˙ H ( q , ˙q ; t ) is invariant with ˙ H ( r , ˙r ; t ) and hence the notion ˙ H -invariance is aproper terminology to be used hereinafter. Such invariance, however, gives us the authority to use the exact solutionsof one system and map it into the other. This would consequently enrich the class of exactly solvable dynamicalsystems within the standard Lagrangian/Hamiltonian settings. Moreover, one should be aware that when equation(35) is multiplied by ˙ q i and summed over i = 1 , , · · · , n , it would yield (43) or equivalently (45). IV. ISOCHRONOUS n -DIMENSIONAL PDM HARMONIC OSCILLATORS: LINEARIZABILITY AND ˙ H -INVARIANCE Having had settled down the technical mathematical issues in the preceding section, we may now proceed to discussthe n -dimensional PDM harmonic oscillators linearizability, ˙ H -invariance and isochronicity.We begin with the n -dimensional PDM oscillator Lagrangian L ( r , ˙r ; t ) = 12 m ( r ) ˙r − V ( q ( r )) = 12 m ( r ) n X j =1 ˙ x j − ω Q ( r ) n X j =1 x j , (47)where the oscillator potential is now assumed to be PDM-deformed in such a way that r −→ p Q ( r ) r as a consequenceof the PDM-deformation of the velocity vector ˙r −→ p m ( r ) ˙r . The substitution of the PDM oscillator Lagrangian(47) in the n Euler-Lagrange equations of motion (27) would result¨ x i + ˙ m ( r ) m ( r ) ˙ x i − m ′ ( r )2 r m ( r ) n X j =1 ˙ x j x i + s Q ( r ) m ( r ) ω x i = 0 , (48)where we have used the relation (38) in the process. On the other hand, the n -dimensional constant mass oscillatorLagrangian L ( q , ˙q ; t ) = 12 m ◦ ˙q − m ◦ ω q = 12 m ◦ n X j =1 ˙ q j − m ◦ ω n X j =1 q j , (49)yields the n Euler-Lagrange linear differential equations (with m ◦ = 1)¨ q i + ω q i = 0 , (50)that admit exact sinusoidal oscillatory solutions q i = A i cos ( ωt + ϕ ) . (51)Using our point transformation of (36)-(39) in (50) one obtains p m ( r ) (cid:20) ¨ x i + ˙ m ( r )2 m ( r ) ˙ x i (cid:21) + p Q ( r ) ω x i = 0 ⇐⇒ ¨ x i + ˙ m ( r )2 m ( r ) ˙ x i + s Q ( r ) m ( r ) ω x i = 0 . (52)This result clearly suggests that, under the current point transformation, the linearizability of (48) into (50) is onlypossible for the one-dimensional case. Whereas, for the n -dimensional case we observe that the invariance could not beestablished and the linearization is not feasible. Nevertheless, the two systems are readily ˙ H invariant. That is, if theexact solutions of one of the systems is known, then we may reflect/map it (through the current point transformation)into the exact solutions of the other system. This would, in effect, authorize the use of the exact solutions (51) of(50) to find the solutions of (48). This is illustrated in the sample of examples below. V. ISOCHRONOUS n -DIMENSIONAL NONLINEAR PDM OSCILLATORS: ILLUSTRATIVEEXAMPLESA. One-dimensional isochronous nonlinear PDM oscillators For the one-dimensional case one should be aware that the dynamical equations in (48) and (52), associated withthe one-dimensional PDM-oscillators Lagrangians (47) L = 12 m ( x ) ˙ x − Q ( x ) ω x , are identical and the Euler-Lagrange invariance is very well established. Moreover, the linearizability of (48) into (50)is possible and straightforward.
1. A PDM without singularity: m ( x ) = 1 / (cid:0) λ x (cid:1) A PDM in the form of m ( x ) = 11 + λ x , (53)would result, by (38), in the coordinate deformation p Q ( x ) = 1 λx ln (cid:16) λx + p λ x (cid:17) . (54)Consequently, the dynamical equation (48), or (52), for the one-dimensional PDM-oscillator Lagrangian L = ˙ x (cid:0) λ x (cid:1) − ω λ ln (cid:16) λx + p λ x (cid:17) ω , (55)in (47) reads ¨ x − λ x λ x ˙ x + p λ x λ ln (cid:16) λx + p λ x (cid:17) ω = 0 , (56)Where its exact solution is inherited from (51) along with (37) so that q = A cos ( ωt + ϕ ) = p Q ( x ) x ⇐⇒ x = 12 λ (cid:16) e λA cos( ωt + ϕ ) − e − λA cos( ωt + ϕ ) (cid:17) , (57)which exactly satisfies (56) and forms its exact isochronous (i.e., ω is amplitude-independent and no constraints areimposed upon it) nonlinear PDM-oscillators solutions, therefore.
2. Two coordinate deformations without/with singularity: Q ( x ) = 1 / (cid:0) ± λ x (cid:1) Position-dependent coordinate deformations in the form of p Q ( x ) = r ± λ x , (58)would imply, by (38), two PDM functions given by m ( x ) = (cid:18) ± λ x (cid:19) . (59)Then the dynamical equation (48), or (52), for the one-dimensional PDM-oscillator Lagrangian (47) L = 12 " ˙ x (cid:0) ± λ x (cid:1) − ω x ± λ x , (60)yields ¨ x ∓ λ x ± λ x ˙ x + (cid:0) ± λ x (cid:1) ω x = 0 , (61)that admits, using (51) and (37), exact solution in the form of q = A cos ( ωt + ϕ ) = p Q ( x ) x ⇐⇒ x = A cos ( ωt + ϕ ) q ± λ A cos ( ωt + ϕ ) , (62)which exactly satisfy the dynamical systems in (61) and hence represent their exact isochronous nonlinear PDM-oscillators solutions.
3. A coordinate deformation with a singularity: Q ( x ) = 1 / (1 − λx ) A coordinate deformation in the form of p Q ( x ) = r − λx , (63)would imply that the PDM function is m ( x ) = −
14 ( λx − ( λx − (64)Using (51) and (37) one obtains q = A cos ( ωt + ϕ ) = r − λx x ⇐⇒ x = A ωt + ϕ ) (cid:20) − λA cos ( ωt + ϕ ) ± q λ A cos ( ωt + ϕ ) + 4 (cid:21) , (65)which satisfies the corresponding dynamical equation, (52),¨ x − λ ( λx − λx −
1) ( λx −
2) ˙ x + 2 ( λx − λx − ω x = 0 , (66)and represents its exact isochronous nonlinear PDM-oscillator solution.
4. A power-law type PDM: m ( x ) ∼ x υ A power-low type coordinate deformation p Q ( x ) = a x υ (67)would result the power-law type PDM function m ( x ) = a ( υ + 1) x υ . (68)Hence, using (51) and (37), the exact isochronous nonlinear PDM-oscillator solution would be q = A cos ( ωt + ϕ ) = a x υ +1 ⇐⇒ x = (cid:20) Aa cos ( ωt + ϕ ) (cid:21) / ( υ +1) , (69)that satisfies the dynamical equation, (52),¨ x + υx ˙ x + 1 υ + 1 ω x = 0 ; υ = − . (70)0
5. An exponential- type PDM: m ( x ) = e λx An exponential-type PDM m ( x ) = e λx (71)would imply, by (38), that the coordinate deformation is p Q ( x ) = e λx λx (cid:0) − e − λx (cid:1) . (72)Which when substituted in the dynamical equation (52) yields¨ x + λ ˙ x + ω λ (cid:0) − e − λx (cid:1) = 0 . (73)Using (51) and (37), one finds that it admits exact isochronous nonlinear PDM-oscillator solution q = A cos ( ωt + ϕ ) = 1 λ (cid:0) − e λx (cid:1) ⇐⇒ x = 1 λ ln (1 − λA cos ( ωt + ϕ )) . (74) B. n -dimensional isochronous nonlinear PDM oscillators For the n -dimensional PDM-oscillators Lagrangian (47) case, we shall recollect that the Euler-Lagrange invariancefalls short and incomplete. One has therefore to appeal to ˙ H -invariance and use the exact solution (51) of (50) toextract exact solutions for (48), where the linearizability of (48) into (50) turned out to be not feasible.
1. Two coordinate deformations without/with singularity: Q ( r ) = 1 / (cid:0) ± λ r (cid:1) The coordinate deformations of the form p Q ( r ) = r ± λ r ; r = vuut n X j =1 x j , (75)would result, by (38), two PDM function m ( r ) = 1 (cid:0) ± λ r (cid:1) . (76)This would allow us to write (37) as q = A cos ( ωt + ϕ ) = p Q ( r ) r ⇐⇒ r = q p ∓ λ q ⇐⇒ x i = A i cos ( ωt + ϕ ) q ∓ λ A cos ( ωt + ϕ ) ; A = vuut n X j =1 A j . (77)which satisfy our dynamical equations of (48)¨ x i ∓ λ ± λ r n X j =1 x j ˙ x j ˙ x i ± λ ± λ r n X j =1 ˙ x j x i + (cid:0) ± λ r (cid:1) ω x i = 0 , (78)and forms their exact n -dimensional isochronous nonlinear PDM-oscillators solutions, therefore.1
2. A power-law type PDM: m ( x ) ∼ r υ Consider a power-law type coordinate deformation p Q ( r ) = a r υ , (79)which in turn implies a PDM function m ( r ) = a ( υ + 1) r υ ; υ = − . (80)Consequently, with q = A cos ( ωt + ϕ ), equation (37) yields q = a r υ r ⇐⇒ r = (cid:18) q − υ a (cid:19) / ( υ +1) q ⇐⇒ x i = A i cos ( ωt + ϕ ) [ A cos ( ωt + ϕ )] − υ a ! / ( υ +1) , (81)as the exact n -dimensional isochronous nonlinear PDM-oscillators solutions for the dynamical equations (48)¨ x i + 2 υr n X j =1 x j ˙ x j ˙ x i − υr n X j =1 ˙ x j x i + ω υ + 1 x i = 0 ; r = n X j =1 x j . (82)In the sample of illustrative example discussed above, we notice that there are no constraints on the frequenciesof the nonlinear PDM oscillators considered. Such frequencies are clearly amplitude-independent and are isochronic.Therefore, all our examples are isochronous nonlinear PDM oscillators. VI. CONCLUDING REMARKS
In this work, we have considered the n -dimensional PDM-Lagrangians in their standard form ( i.e., the differencebetween kinetic and potential energies). However, in order to make our study comprehensive and self-contained,we have recollected and elaborated on the solvability (exact and conditionally exact) and linearizability of the non-standard Mathews-Lakshmanan nonlinear oscillators (2). The generalization of such nonlinear oscillators (2) to anyPDM, m ( r ), settings is also discussed and reported in section II. Yet we have asserted that the position-dependentdeformation of time (manifested by the nonlocal point transformations in (9) or (12)) renders such PDM nonlinearoscillators non-isochronous so that their frequencies become amplitude-dependent.To preserve isochronicity of the PDM nonlinear oscillators, we had to return back to the standard Lagrangians formto obtain an interesting sets of isochronous PDM nonlinear oscillators. In so doing, we have shown/emphasized (insection III) that negative the gradient of the PDM potential force field (i.e., the force vector associated with PDMsettings) is no longer given by the time derivative of the canonical momentum, p ( r ) = m ( r ) ˙r , but it is rather givenin terms of the pseudo-momentum, π ( r ) = p m ( r ) ˙r [5, 7] (or the Noether momentum as in [6]). That is, − ∇ V ( q ( r )) = F = p m ( r ) ddt (cid:16)p m ( r ) ˙r (cid:17) , where q ( r ) = p Q ( r ) r , with m ( r ) and Q ( r ) satisfy the correlation p m ( r ) = p Q ( r ) (cid:18) Q ′ ( r )2 Q ( r ) r (cid:19) . In the same section, moreover, we have shown that the connection between constant mass settings and PDM settingsis feasible through some point transformation, where the time is kept as is (i.e., no position-dependent deformation oftime). Hereby, the Euler-Lagrange invariance is shown satisfactory for n = 1 but unsatisfactory/incomplete for n ≥ H -invariance (where ˙ H = dH/dt ). Moreover, such invariance goes alongside with the fact that the totalenergy is a conserved quantity (documented in (46)) and is a constant of motion (i.e., integral of motion), therefore.This result allowed us to use, in section IV and V, the well known exact solutions (51) of the linear oscillator (50)along with our point transformation (37) to obtain exact solutions for a set of n -dimensional isochronous nonlinear2PDM oscillators. This is documented in the illustrative examples of section V, where a set of one-dimensional and aset of n -dimensional isochronous nonlinear PDM oscillators are reported.In the light of our experience in the current methodical proposal, we argue that the linearizability of the equationsof motion (48) of the standard PDM oscillators Lagrangians (47) is only possible for the one-dimensional systems.Whereas, for the n -dimensional case we observe that the invariance could not be established and the linearization isnot feasible (documented in (48) to (52)). Nevertheless, the constant mass and the PDM systems ( (49) and (47),respectively) are readily ˙ H invariant (reported in (34) to (46)). This would, in effect, authorize the use of the exactsolutions (51) of (50) to find the solutions of (48). In a more general language, if the exact solution of the constantmass system in (34) is known, then we may reflect/map it (through the current point transformation) into the solutionof the corresponding PDM system in (25) (the other way around is also true, of course). To the best of our knowledge,such results and/or methodical proposal have never been reported elsewhere in the literature.3 [1] P. M. Mathews, M. Lakshmanan, Quart. Appl. Math. (1974) 215.[2] A. K. Tiwari, S. N. Pandey, M. Santhilvelan, M. Lakshmanan, J. Math. Phys. (2013) 053506.[3] M. Lakshmanan, V. K. Chandrasekar, Eur. Phys. J. Special Topics (2013) 665.[4] R.G. Pradeep, V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, J. Math. Phys. (2009) 052901.[5] O. Mustafa, J. Phys. A ; Math. Theor. (2015) 225206.[6] J. F. Cari˜nena, M. F. Ra˜nada, M. Santander, J. Phys. A : Math. Theor. (2017) 465202.[7] J. F. Cari˜nena, M. F. Ra˜nada, M. Santander, M. Senthilvelan, Nonlinearity (2004) 1941.[8] A. Bhuvaneswari, V. K. Chandrasekar, M. Santhilvelan, M. Lakshmanan, J. Math. Phys. (2012) 073504.[9] M. Lakshmanan, S. Rajasekar, Nonlinear Dynamics: Integrability, Chaos, and Patterns (Springer-Verlag, Berlin, 2003).[10] I. Boussaada, A. R. Chouikha, J. M. Strelcyn, Bull. Sci. Math. (2011) 89.[11] M. Bardet, I. Boussaada, A. R. Chouikha, J. M. Strelcyn, Bull. Sci. Math. (2011) 230.[12] O. Von Roos, Phys. Rev.
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